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  • Letter
  • Open Access

Directed extended-range percolation

Wenbo Liu1, Yiwen Zeng1, Xueming Liu1,*, and Ginestra Bianconi2,†

  • *Contact author: xm_liu@hust.edu.cn
  • †Contact author: ginestra.bianconi@gmail.com
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Phys. Rev. E 114, L042301 – Published 26 October, 2026

DOI: https://doi.org/10.1103/hs2n-6rsn

Abstract

For standard percolation, directionality is known to increase the complexity of percolation, requiring the distinct statistical treatment of in messages and out messages for determining the percolation order parameters. Here we show that when connectivity is ensured by paths of length R≥2, network directionality, impeding backtracking, can significantly reduce the complexity of percolation. To illustrate this finding, we introduce directed extended-range percolation (DERP), defined on directed networks with nonreciprocal edges and motivated by applications in quantum communication. In this framework, message transmission is enabled between trusted nodes separated by a directed path of length at most R. Using a message-passing approach, we show that, on locally treelike structures, directionality enables an exact determination of the percolation threshold and the anomalous critical indices with a number of distinct statistical messages smaller than the naive expectation, which would require twice as many messages as in the undirected version. This reduction arises because directionality suppresses backtracking paths, thereby considerably simplifying the resulting equations. On random directed networks we find that the critical behavior of DERP depends sensitively on degree correlations. These analytical predictions are corroborated by extensive Monte Carlo simulations, highlighting the profound impact of directionality and correlations on long-range connectivity in complex networks.

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Phys. Rev. E 114, L042301– Published 26 October, 2026
  • Received 23 May 2026
  • Accepted 8 October 2026
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